L(s) = 1 | − 2-s + 4-s − 2·5-s + 7-s − 8-s − 3·9-s + 2·10-s − 4·11-s − 14-s + 16-s − 6·17-s + 3·18-s − 2·20-s + 4·22-s + 8·23-s − 25-s + 28-s − 10·29-s + 8·31-s − 32-s + 6·34-s − 2·35-s − 3·36-s − 6·37-s + 2·40-s + 6·41-s + 4·43-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.377·7-s − 0.353·8-s − 9-s + 0.632·10-s − 1.20·11-s − 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.707·18-s − 0.447·20-s + 0.852·22-s + 1.66·23-s − 1/5·25-s + 0.188·28-s − 1.85·29-s + 1.43·31-s − 0.176·32-s + 1.02·34-s − 0.338·35-s − 1/2·36-s − 0.986·37-s + 0.316·40-s + 0.937·41-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5981342555\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5981342555\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.791683393935935773346976116547, −8.310993556895860538285090281250, −7.55111798933411457737769636175, −6.98546602109746736864670767573, −5.85357656265532644668678830790, −5.09668062546972176782846052974, −4.10388224753287373041649125168, −2.96897904490708863142085706195, −2.21452719189805902343262068063, −0.52137795477594883297470319142,
0.52137795477594883297470319142, 2.21452719189805902343262068063, 2.96897904490708863142085706195, 4.10388224753287373041649125168, 5.09668062546972176782846052974, 5.85357656265532644668678830790, 6.98546602109746736864670767573, 7.55111798933411457737769636175, 8.310993556895860538285090281250, 8.791683393935935773346976116547